Question

Prove that 3+2√5 is irrational​

Answer 1

Step-by-step explanation:

To prove 3+2√5 is irrational.

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Let us assume that 3+2√5 is a rational number and let it's simplest form is P/Q.

then, p and q are integers having no common factor other than 1 and q ≠0.

Then,

p/q =3+2√5

P/q–3 = 2√5

(p–3q)/q = 2√5

Since, P and q are integers having no common factor other than 1 and q ≠0.

So,(p–3q)/q is a rational number.

Thus, 2√5 is also a rational number.

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But, this contradicts the fact that 2√5 is a irrational number .

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The contradiction arises by assuming 3+2√5 is rational.

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so, our assumption is wrong.

Hence, 3+2√5 is irrational.

proved.